Annotation of the results obtained in 2021
1. One of the fundamental problems in mathematical physics is the black hole information loss problem. An approach to solving this problem was recently proposed on the basis of the so-called "island formula" for the entanglement entropy of Hawking radiation. Although this recipe was proposed within the holographic approach, the island rule has been supposed to apply to black holes in more general theories. Black holes in a four-dimensional asymptotically flat space-time were considered in [Hashimoto et al., arXiv: 2004.05863] and others, there the derivation of the Page curve was proposed. However, in the work of the project participants, this behavior was refuted; for an evaporating black hole, the entropy of island configurations rapidly increases with time as the mass is approached a critical value. In the project is also shown, that if we consider this entropy from the point of view of a certain accelerated system then the growth of entropy is suppressed. 2. A general solution depending on two constants and one arbitrary function of the Bogomol’nyi equations for the SO(3) gauge model with the triplet of scalar fields in the fundamental representation is found. It is shown that the solution yields the Bogomol’nyi-Prasad-Zommerfield and Singleton solutions as particular cases. The distribution of spins in elastic media corresponding to the Bogomol’nyi-Prasad-Zommerfield solution is computed in the framework of the geometric theory of defects. The global existence theorem for the conformal gauge is proved for a Riemannian metric given on topologically trivial two-dimensional manifold. 3. Investigation of operation of genome as a functional program and biological evolution as learning for functional programs was performed. Gene regulation was discussed as monadic computation in functional programming. This supports the idea of genome as a program written in Haskell-like programming language where recursive applications of lists of functions (genes) as applicative functor express parallel processes in a cell and gene regulation can be described by monadic computations. 4. A model of quantum dynamics of density matrix with interactions between populations and coherencies and dissipation where dissipation and decoherence depend on the state of the system. This dependence is described by a model of quantum control. Current for quantum transfer in this model will be directed (i.e. the direct and the reverse currents for the dynamics under consideration will be different). 5. Memory models for open quantum system dynamics where the memory exponentially decays with time were investigated. In addition, models with memory for cold reservoirs were considered. 6. Limit theorems for compositions of independent random linear operators in a Banach space, which are non-commutative analogs of the law of large numbers and the central limit theorem, are proved. Properties of random walks in the position space and momentum space of quantum system with infinite-dimensional position space are described. Generators of semigroups describing mean values of functionals on random walks are obtained. These semigroups describe the solution of diffusion equation with self-adjoint Laplace operator of type Gross-Volterra. 7. The Levy Laplacian is an infinite-dimensional Laplacian that has no finite-dimensional analog. The interest in this operator is due to its connection with the Yang-Mills equations. The family of modified Levy Laplacians is generated by the action of infinite-dimensional rotations on the Levy Laplacian. The geometry conditions on a 4-dimensional manifold were found, which imply that a solution of the Laplace equation for the modified Levy Laplacian on the manifold of paths corresponds to an instanton on the 4-dimensional manifold.

 

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Annotation of the results obtained in 2019
1. We have shown that one common formulation of Eigenstate thermalization hypothesis (ETH) does not necessarily imply thermalization (in some sense) of an observable of isolated many body quantum systems. To get such thermalization one has to postulate it in the ETH-ansatz. There are vector states of the special form such that the expectation value of an observable in these states is approximately equal to the canonical thermal average. 2. Random semigroup is defined as the measurable mapping of the space with the measure into the locally convex space of strongly continuous operator-valued function on real semiaxe such that the values of this mapping are semigroups. The mean value of a random semigroup is point-vice defined by the Pettis integral. But the mean value of random semigroup can has no semigroup property. The procedure of generalized averaging of random semigroup is constructed by means of she applying of the Chernoff theorem for the sequence of independent identically distributed random semigroups and its compositions. This procedure of averaging of random semigroup preserves the semigroup property for the mean value. The constructed approach to the investigation of random operators is applied to the studying of operators of random shifts on the vectors of infinite dimension Banach space. This objects are useful both in the studying of random process with values in Banach spaces and in the studying of a regularization of singular boundary value problems.To this aim we study the measures on the Banach spaces which is invariant with respect to a shift on arbitrary vector. We obtain the necessary and sufficient conditions of strong continuity for the mean value of random shifts semigroup. The description of the self-adjoined contraction semigroups in Hilbert space is obtained. 3. We describe the stochastic limit for quantum particle with non-linear interaction with quantum Bose field. We get expressions for commutation relations for collective degrees of freedom of the field and the particle in the stochastic limit (or the master field) and show that the statistics for these collective degrees of freedom becomes free (or quantum Boltzmann). We describe an analogue of the temperature double construction in the free case. In this case the two fields of the temperature double in this model commutation relations will be different. We construct the free Fock module for the quantum temperature double construction used in this model. 4. The basic model for many physical systems is a set of harmonic oscillators. In this work we consider such a set interacting with an environment. The dynamics of a small number of oscillators interacting with the reservoir which quickly relaxes to its equilibrium state (it corresponds to the so-called Markovian approximation) is well studied and has wide applications in physics. However, the modern problems require consideration of an arbitrary number of oscillators. Moreover, they lead to reservoirs of a more general form which "remember" the history of interaction with the system (i.e. the interaction with it goes beyond the Markovian approximation). Description of the dynamics of such systems has been done in this paper. 5. The description of the continuous distribution of spins in a three-dimensional medium (for example, in a ferromagnet) in the traditional approach is carried out using the n-field, which is the main variable in the models of such media. The singularities of the n-field are called disclinations. In the presence of one or several disclinations, the traditional approach can be used if the corresponding boundary conditions are set for the n-field. In the case of a large number of disclinations, the boundary conditions are significantly complicated, and the solution of the corresponding problems becomes extremely complicated. In the limit case of a continuous defect distribution, the traditional approach becomes inapplicable, since the n-field has singularities at each point and, therefore, does not exist. To solve this problem, a new variable is introduced in the geometric theory of defects — the SO (3)-connection, which has significantly fewer singularities in the presence of disclinations than the n-field. In those areas of the medium where there are no disclinations, the curvature is zero, the SO (3)-connection is a pure gauge field and is expressed through the partial derivatives of the field of rotation angles. The change of n-field by the rotational angle field in the geometric theory of defects implies the appearance of the additional degree of freedom. This degree of freedom is shown to be gauge and corresponding to local SO(2)-rotations around n-field. We prove that an arbitrary expression for the free energy written in terms of the n-field results in the equivalent gauge model which is invariant under local SO(2)-transformations but does not contain a gauge field. Explicit form of the gauge transformations and the corresponding linear approximation are found. As an example, the gauge model of Heisenberg’s ferromagnet which has straightforward generalization in the geometric theory of defects is constructed. 6. A covariant definition of the Levy Laplacian (a definition which is not depended of the choice of the coordinate map on an infinite dimensional manifold) is introduced. The relation between the quasi-linear Yang–Mills heat equations on a finite-dimensional manifold and the linear heat equation for the Levy Laplacian on an infinite-dimensional manifold is found. Namely, Accardi–Gibilisco–Volovich theorem on the equivalence of the Yang–Mills equations and the Laplace equation for the Levy Laplacian is generalized in the following way. It is shown that a time-depended connection in a finite-dimensional vector bundle is a solution of the Yang–Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the Levy Laplacian. 7. Within the framework of this project, two-dimensional quantum gravity of Jackiw-Teitelboim (JT) has studied, namely, the study of nonperturbative effects associated with contributions from two-dimensional geometries with non-trivial topologies has been performed. Using the matrix representation for JT gravity, the behavior of universes with a complex topological structure, the so-called "Baby universes" has been studied. JT gravity is closely related to quantum mechanical fermion model of Sachdev-Ye-Kitayev. For the 0-dimensional version of this model, the presence of off-diagonal quasi-avereging in the limit of large N has been shown. This effect is an analogue of spontaneous symmetry breaking in the Higgs model, which underlies the Standard Model of elementary particles. 8. The studies of two-dimensional quantum gravity were conducted. The main tool in these studies is the quantum-mechanical SYK model, which is holographically dual to a certain two-dimensional gravity theory. The key aspect is the study of nonperturbative effects in the gravitational theory, in particular of contributions from two-dimensional geometries with nontrivial topology. From the SYK perspective, these effects correspond to nontrivial saddle points of the 1/N expansion of the model. Within the present project the new analytic solutions are obtained, which describe these effects in the SYK model in the limit of large order of interaction. An important question is construction of observables in the gravity theory, which have direct interpretation in SYK terms and are sensitive to the nonperturbative effects. The holographic two-point correlation functions of matter fields on the background of two-dimensional geometry with nontrivial topology were studied as a primary candidate for the role of such observables.

 

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Annotation of the results obtained in 2020
1. It is shown that with the help of the analog of the Bogolyubov–van Hovу scaling conditions used in the stochastic limit theory one can solve the large temperature and radiative energy increases of Hawking vaporizing black holes for the cases of Kerr and Reissner–Nordstrom. An analogue of the scaling conditions implies a special relation of mass and parameters of the listed black holes. 2. The classes functions of Sobolev spaces and the classes of smooth functions on the real separable Hilbert space endowed with the shift-invariant measure are introduced. The conditions of embedding and dense embedding of the space of smooth functions into the Sobolev space are obtained. If this condition violated then the example of the Lavrent’ev effect is observed. It means that the minimum of the functional of Puasson-Dirichlet problem on the space of smooth function is differ from the minimum of this functional on the Sobolev space. The convergence of the sequence of compositions of independent random operator valued processes is studied. The evolution equation corresponded to the limit process is obtained. 3. The statistics for collective degrees of freedom for non-linear interaction of quantum field with quantum particle is studied. We show that in the weak coupling limit these collective degrees of freedom satisfy some variant of free statistics. A model of biological evolution as a model of learning theory is considered. Relation of universal scaling in genomics and universal regularization in learning model under consideration of the form of estimate for Kolmogorov complexity (as in the model by Yu.I.Manin for the Zipf’s law «complexity as energy»). 4. The Chern-Simons action is used for the description of point disclinations in the geometric theory of defects. The most general spherically symmetric flat SO(3)-connection and corresponding distribution of n-field are found. Two examples of point disclinations are described. 5. The Levy Laplacian is the Cesaro mean of n-dimensional Laplacians as n tends to infinity. Previously, it was known about the equivalence of the Yang-Mills equations and the Laplace equation for the Levy Laplacian for the parallel transport along a contour. The Lévy Laplacian is not invariant under the action of the rotation group. This action generalizes a family of modified Levy Laplacians. It is proved that a connection is an instanton if and only if the parallel transport generated by the connection is a solution of a system consisting of three Laplace equations for modified Lévy Laplacians. 6. Further research of quantum dynamics models of systems with memory is being done. It turns out that some of such models correctly describe the dynamics of an open system in an arbitrary time range. This is due to the fact that these models are approximate and the assumptions they contain may be broken at certain times. It has been shown that the requirement for these models correctness in arbitrary time ranges leads to quite definite limitations on coefficients of equations arising in the models under consideration. 7. Within the present project the studies of two-dimensional quantum gravity were conducted. The main tool in these studies is the quantum-mechanical Sachdev–Ye–Kitaev (SYK) model, which is holographically dual to a certain two-dimensional gravity theory. One of the criteria of this duality is the fact that the SYK model is a maximally chaotic quantum system. In the present project the physical sense of the nonperturbative effects obtained before has been studied in the context of quantum chaos. 8. The problem of giving a rogorous sense to the so-called microscopic solutions of the Boltzmann–Enskog kinetic equation discovered by N.N. Bogolubov, which have the form of sums of delta functions and correspond to the reversible dynamics of hard spheres, has been considered. However, direct substitution of such solutions into the equation leads to formally indefinite products of delta functions. Necessary regularizations were proposed in the previous works of one of the project participants. In the reporting year another method of regularization of the collision integral of the Boltzmann–Enskog equation was proposed, which requires neither regularizations of the delta functions themselves nor representation of solutions of the Boltzmann–Enskog equations in the form of a perturbation theory series (which was necessary in previous works). Thus, the simplest method of making rogorous sense to these solutions has been proposed. The significance of this result for the general topic of the grant related to the problem of irreversibility is that these solutions reconcile, in a sense, reversible and irreversible dynamics within the same equation. Namely, reversibility or irreversibility depends on the class of solutions considered: reversibility and entropy growth in the case of regular solutions and irreversibility in the case of singular microscopic solutions.

 

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